An elementary proof and detailed investigation of the bulk-boundary correspondence in the generic two-band model of Chern insulators

Abstract

With the inclusion of arbitrary long-range hopping and (pseudo)spin-orbit coupling amplitudes, we formulate a generic model that can describe any two-dimensional two-band bulk insulators, thus providing a simple framework to investigate arbitrary adiabatic deformations upon the systems of any arbitrary Chern numbers. Without appealing to advanced techniques beyond the standard methods of solving linear difference equations and applying Cauchy's integral formula, we obtain a mathematically elementary yet rigorous proof of the bulk-boundary correspondence on a strip, which is robust against any adiabatic deformations upon the bulk Hamiltonian and any uniform edge perturbation along the edges. The elementary approach not only is more transparent about the underlying physics but also reveals various intriguing nontopological features of Chern insulators that have remained unnoticed or unclear so far. Particularly, if a certain condition is satisfied (as in most renowned models), the loci of edge bands in the energy spectrum and their (pseudo)spin polarizations can be largely inferred from the bulk Hamiltonian alone without invoking any numerical computation for the energy spectrum of a strip.

arXiv comment55 pages, 18 figures; follow-up to arXiv:1705.06913, some material thereof repeated; v3: various minor improvements made, version to appear in IJMPB

References in corpus (17)

• Topological Insulators
• Topological insulators and superconductors
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Periodic table for topological insulators and superconductors
• Anomalous edge state in a non-Hermitian lattice
• Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors
• Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems
• Topological Transition in a Non-Hermitian Quantum Walk
• Bulk-boundary correspondence of topological insulators from their Green's functions
• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• Bulk-edge correspondence for two-dimensional topological insulators
• Bulk-boundary correspondence from the inter-cellular Zak phase
• Bulk-Edge Correspondence in 2+1-Dimensional Abelian Topological Phases
• Shockley model description of surface states in topological insulators
• Unified bulk-boundary correspondence for band insulators
• Bulk-edge correspondence in graphene with/without magnetic field: Chiral symmetry, Dirac fermions and Edge states
• An elementary rigorous proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model